Global continuation of forced oscillations of retarded motion equations on manifolds Pierluigi Benevieri, Alessandro Calamai, Massimo Furi and Maria Patrizia Pera To the outstanding mathematician Andrzej Granas whose contribution to the fixed point index theory made the existence of this article possible Abstract. We investigate T -periodic parametrized retarded functional motion equations on possibly noncompact manifolds; that is, constrained second order retarded functional differential equations. For such equations we prove a global continuation result for T -periodic solutions. The approach is topological and is based on the degree theory for tangent vector fields as well as on the fixed point index theory. Our main theorem is a generalization to the case of retarded equations of an analogous result obtained by the last two authors for second order differential equations on manifolds. As corollaries we derive a Rabinowitz-type global bifurcation result and a Mawhin-type continuation principle. Finally, we deduce the existence of forced oscillations for the retarded spherical pendulum under general assumptions. Mathematics Subject Classification. 34K13, 70K40, 34C40, 37C25. Keywords. Retarded functional differential equations, global bifurcation, periodic solutions, fixed point index theory, forced motion on manifolds. 1. Introduction Let M R k be a possibly noncompact boundaryless smooth manifold, and denote by BU, 0], M the metric space of bounded and uniformly continuous maps from, 0] into M with the topology of the uniform convergence. Consider the following parametrized retarded functional motion equation on M: x πt = λft, x t, 1.1 where x πt stands for the tangential part of the acceleration x t R k at the point xt M,
2 P. Benevieri, A. Calamai, M. Furi and M. P. Pera λ is a nonnegative real parameter, f : R BU, 0], M R k is such that ft, φ T φ0 M for all t, φ, where T p M R k is the tangent space of M at a point p of M. We call a continuous map f : R BU, 0], M R k verifying the above tangency condition a functional field. In addition, we use the standard notation in functional equations: whenever it makes sense, x t BU, 0], M denotes the function θ xt + θ. Roughly speaking, a retarded functional motion equation on M is a motion equation constrained on M in which the active force, λf in this case, takes into account the whole past history of the process. In this paper we are interested in obtaining global continuation results for 1.1. That is, we assume that the functional field f is T -periodic in the first variable and locally Lipschitz in the second one, and we study the topological properties of the set of T -periodic solutions of 1.1. More in detail, we denote by CT 1 M the space of the T -periodic C1 maps x: R M with the standard C 1 metric, and we call λ, x [0, + CT 1 M a T -forced pair of equation 1.1 if x: R M is a T -periodic solution of 1.1 corresponding to λ. Among these pairs we distinguish the trivial ones; that is, the elements of the form 0, p, where, given p M, p denotes the constant p-valued function defined on R. We call a point p M a bifurcation point of 1.1 if any neighborhood of 0, p in [0, + CT 1 M contains nontrivial T -forced pairs. The main result of this paper, Theorem 4.4, is a global continuation result for T -forced pairs of equation 1.1. That is, given an open subset Ω of [0, + CT 1 M, Theorem 4.4 provides sufficient conditions for the existence of a global bifurcating branch in Ω, meaning a connected subset of Ω of nontrivial T -forced pairs whose closure in Ω is noncompact and intersects the set of trivial T -forced pairs. The prelude of our approach can be found in some papers of the last two authors see, e.g., [9, 10]. In [10] they proved a global continuation result for the second order parametrized motion equation x πt = λft, xt, x t, λ 0, 1.2 where f: R T M R k is a continuous, T -periodic tangent vector field on M. Such a result is obtained by means of topological methods. In particular, the existence of a global bifurcating branch is given if the degree in an open subset of M of the tangent vector field the average force fp = 1 T T 0 ft, p, 0 dt, p M, is defined and nonzero. Our results improve those of [10] in a natural sense, since differential equations with delay include ordinary differential equations ODEs as particular cases. On the other hand, the extension that we obtain is only partial for the following reasons. First, in 1.2 the active force may depend also on the velocity which is not the case, in the present setting,
Global continuation of forced oscillations 3 for 1.1. Second, for technical reasons we are led to assume that the functional field f is locally Lipschitz in the second variable, so we are not able to prove our results with the sole continuity assumption as it was done in [10]. The proof of Theorem 4.4 is obtained proceeding in the spirit of [10]. Such a proof is based on a relation, proved in a previous paper by the authors [3, Lemma 3.8], between the degree in an open subset of M of the tangent vector field fp = 1 T T 0 ft, p dt, p M, and the fixed point index of a sort of Poincaré T -translation operator acting inside the Banach space C[ T, 0], R 2k. Notice that 1.1 is equivalent to a first order retarded functional differential equation RFDE on the tangent bundle T M. We prove Theorem 4.4 by transforming 1.1 into the first order system { x t = yt, y 1.3 t = rxt, yt + λft, x t, λ 0, where r : T M R k is the inertial reaction. On the other hand, observing the above system, one can see that a bifurcation problem associated with 1.1 cannot be simply reduced to a RFDE on T M of the form z t = λht, z t and system 1.3 is not in this form, since the inertial motion problem does not correspond, in the phase space, to the trivial equation z t = 0. Recently, we have devoted some papers to the study of first and second order RFDEs on differentiable manifolds. The problem that we address in the present paper is characterized by a broad generality and presents some peculiar features which make it very hard and challenging. In a sense, our previous papers turn out to be a preliminary investigation that enable us to better understand how to tackle the problem. In the recent paper [3] we proved a global continuation result for periodic solutions of first order parametrized RFDEs on manifolds. Despite the similarity, as pointed out above, the problem that we address here cannot be handled with the techniques developed in [3] and require a specific study. On the other hand, in [4] we obtained a global continuation result under the assumption that the manifold M is compact. Here we present a consequence of our main result Corollary 4.12 improving the one in [4]. Further, the simultaneous presence of infinite delay and noncompactness of the manifold leads us to work in the space BU, 0], M. In [2] we studied general properties of RFDEs with infinite delay on possibly noncompact differentiable manifolds since we were not aware of the presence of such results in the literature. In contrast, the different and related cases of RFDEs with finite delay in Euclidean spaces have been investigated by many authors. For general reference we suggest the monograph by Hale and Verduyn Lunel [17]. We refer the reader also to the works of Gaines and Mawhin [12], Nussbaum [26,
4 P. Benevieri, A. Calamai, M. Furi and M. P. Pera 27] and Mallet-Paret, Nussbaum and Paraskevopoulos [21]. For RFDEs with infinite delay in Euclidean spaces we recommend the article of Hale and Kato [16], the book by Hino, Murakami and Naito [18], and the more recent paper of Oliva and Rocha [30]. For RFDEs with finite delay on manifolds we suggest the papers of Oliva [28, 29]. Finally, for RFDEs with infinite delay on manifolds we cite the already mentioned paper [2]. We conclude the paper with some consequences of Theorem 4.4. One is a Rabinowitz-type global bifurcation result see [31], obtained by assuming that the degree of the tangent vector field f is nonzero on an open subset of M. Another corollary is a Mawhin-type continuation principle [22, 23] which is a partial extension of an analogous result, for a motion differential equation on M, obtained by the last two authors in [10, Corollary 2.1]. A principal motivation in [10] was to investigate the problem about the existence of forced oscillation for constrained systems with compact, topologically nontrivial constraint, such as the spherical pendulum, whose constraint M is S 2. Concerning the retarded spherical pendulum, an existence result for forced oscillations has been proved by the authors in [1], assuming the continuity of the functional field f on the space R C, 0], S 2, with C, 0], S 2 having the too weak topology of the uniform convergence on compact intervals, making the continuity assumption of f a heavy hypothesis. Our main result allows us to extend this existence theorem to the case in which the functional field is continuous on R BU, 0], S 2. 2. Preliminaries 2.1. Fixed point index We recall that a metrizable space X is an absolute neighborhood retract ANR if, whenever it is homeomorphically embedded as a closed subset C of a metric space Y, there exist an open neighborhood V of C in Y and a retraction r : V C see, e.g., [5, 14]. Polyhedra and differentiable manifolds are examples of ANRs. Let us also recall that a continuous map between topological spaces is called locally compact if each point in its domain has a neighborhood whose image is contained in a compact set. Let X be a metric ANR and consider a locally compact continuous X -valued map k defined on a subset Dk of X. Given an open subset U of X contained in Dk, if the set of fixed points of k in U is compact, the pair k, U is called admissible. We point out that such a condition is clearly satisfied if U Dk, ku is compact and kp p for all p in the boundary of U. To any admissible pair k, U one can associate an integer ind X k, U the fixed point index of k in U which satisfies properties analogous to those of the classical Leray Schauder degree [20]. The reader can see for instance [6, 13, 25, 27] for a comprehensive presentation of the index theory for ANRs. As regards the connection with the homology theory we refer the reader to standard algebraic topology textbooks e.g., [7, 32].
Global continuation of forced oscillations 5 We summarize below the main properties of the fixed point index. Existence: If ind X k, U 0, then k admits at least one fixed point in U. Normalization: If X is compact, then ind X k, X = Λk, where Λk denotes the Lefschetz number of k. Additivity: Given two disjoint open subsets U 1, U 2 of U, if any fixed point of k in U is contained in U 1 U 2, then ind X k, U = ind X k, U 1 + ind X k, U 2. Excision: Given an open subset U 1 of U, if k has no fixed points in U\U 1, then ind X k, U = ind X k, U 1. Commutativity: Let X and Y be metric ANRs. Suppose that U and V are open subsets of X and Y, respectively, and that k : U Y and h: V X are locally compact maps. Assume that the set of fixed points of either hk in k 1 V or kh in h 1 U is compact. Then the other set is compact as well and ind X hk, k 1 V = ind Y kh, h 1 U. Generalized homotopy invariance: Let I be a compact real interval and W an open subset of X I. For any λ I, denote W λ = {x X : x, λ W }. Let H : W X be a locally compact map such that the set {x, λ W : Hx, λ = x} is compact. Then ind X H, λ, W λ is independent of λ. 2.2. Degree of a vector field Let us recall some basic notions on degree theory for tangent vector fields on differentiable manifolds. Let w : M R k be a continuous autonomous tangent vector field on a smooth manifold M, and let V be an open subset of M. We say that the pair w, V is admissible or, equivalently, that w is admissible in V if w 1 0 V is compact. In this case one can assign to the pair w, V an integer, degw, V, called the degree or Euler characteristic, or rotation of the tangent vector field w in V which, roughly speaking, counts algebraically the number of zeros of w in V for general references see, e.g., [15, 19, 24, 33]. Notice that the condition for w 1 0 V to be compact is clearly satisfied if V is a relatively compact open subset of M and wp 0 for all p in the boundary of V. As a consequence of the Poincaré Hopf theorem, when M is compact, degw, M equals χm, the Euler Poincaré characteristic of M. In the particular case when V is an open subset of R k, degw, V is just the classical Brouwer degree of w in V when the map w is regarded as a vector field; namely, the degree degw, V, 0 of w in V with target value 0 R k. All the standard properties of the Brouwer degree in the flat case, such as homotopy invariance, excision, additivity, existence, still hold in the more general context of differentiable manifolds. To see this, one can use an
6 P. Benevieri, A. Calamai, M. Furi and M. P. Pera equivalent definition of degree of a tangent vector field based on the fixed point index theory as presented in [9, 10]. Let us stress that, actually, in [9, 10] the definition of degree of a tangent vector field on M is given in terms of the fixed point index of a Poincaré-type translation operator associated with a suitable ODE on M. We point out that no orientability of M is required for degw, V to be defined. This highlights the fact that the extension of the Brouwer degree for tangent vector fields in the nonflat case does not coincide with the one regarding maps between oriented manifolds with a given target value as illustrated, for example, in [19, 24]. This dichotomy of the notion of degree in the nonflat situation is not evident in R k : it is masked by the fact that an equation of the type fx = y can be written as fx y = 0. Anyhow, in the context of RFDEs on manifolds ODEs included, it is the degree of a vector field that plays a significative role. Moreover, if w has an isolated zero p and V is an isolating open neighborhood of p, then degw, V is called the index of w at p or, by abuse of terminology, the index of p. The excision property ensures that this is a well-defined integer. We close this section with two results that will be used in what follows. Next one is a Wyburn s type topological lemma. Lemma 2.1 See [10]. Let K be a compact subset of a locally compact metric space Y. Assume that any compact subset of Y containing K has nonempty boundary. Then Y\K contains a connected set whose closure is noncompact and intersects K. The following lemma, whose elementary proof is given for the sake of completeness, will be used in the proofs of Theorem 4.3 and Lemma 4.7. Lemma 2.2. Let F : X Y be a continuous map between metric spaces and let {γ n } be a sequence of continuous functions from a compact interval [a, b] or, more generally, from a compact space into X. If {γ n s} converges to γs uniformly for s [a, b], then also F γ n s F γs uniformly for s [a, b]. Proof. Notice that, if C is a compact subset of X, then for any ε > 0 there exists δ > 0 such that x X, c C, dist X x, c < δ imply that dist Y F x, F c < ε. Now, our assertion follows immediately by taking the compact C to be the image of the limit function γ : [a, b] X. 3. Retarded functional differential equations Given an arbitrary subset A of R l, we denote by BU, 0], A the set of bounded and uniformly continuous maps from, 0] into A. For brevity, we will use the notation à := BU, 0], A.
Global continuation of forced oscillations 7 Notice that R l is a Banach space, being closed in the space BC, 0], R l of the bounded and continuous functions from, 0] into R l endowed with the standard supremum norm. Given T > 0, let  := C[ T, 0], A denote the metric subspace of C[ T, 0], R l of the A-valued continuous functions on [ T, 0] and set  := { η  : η T = η0}. We observe that, if A is locally compact, then  but not à is locally complete. Moreover,  and à are complete if and only if A is closed. Throughout the paper, unless otherwise stated, the norm in any Banach space will be denoted by if the dimension of the space is finite and by otherwise. Thus, the distance between two elements γ and δ of à will be denoted by γ δ, even when γ δ does not belong to Ã. Let N be a boundaryless smooth manifold in R l and, given q N, let T q N R l denote the tangent space of N at q. A continuous map g : R Ñ Rl is said to be a retarded functional tangent vector field over N if gt, η T η0 N for all t, η R Ñ. In what follows, any map with this property will be briefly called a functional field over N. Let us consider a first order retarded functional differential equation RFDE of the type z t = gt, z t, 3.1 where g : R Ñ Rl is a functional field over N. Here, as usual and whenever it makes sense, given t R, by z t Ñ we mean the function θ zt + θ. A solution of 3.1 is a function z : J N, defined on an open real interval J with inf J =, bounded and uniformly continuous on any closed halfline, b] J, and which verifies eventually the equality z t = gt, z t. That is, z : J N is a solution of 3.1 if z t Ñ for all t J and there exists τ J such that z is C 1 on the interval τ, sup J and z t = gt, z t for all t τ, sup J. Observe that the derivative of a solution z may not exist at t = τ. However, the right derivative D + zτ of z at τ always exists and is equal to gτ, z τ. Also, notice that t z t is a continuous curve in Ñ, since z is uniformly continuous on any closed half-line, b] of J. A solution of 3.1 is said to be maximal if it is not a proper restriction of another solution. As in the case of ODEs, Zorn s lemma implies that any solution is the restriction of a maximal solution. Given ω Ñ, let us associate with 3.1 the initial value problem { z t = gt, z t, 3.2 z 0 = ω.
8 P. Benevieri, A. Calamai, M. Furi and M. P. Pera A solution of 3.2 is a solution z : J N of 3.1 such that sup J > 0, z t = gt, z t for t > 0, and z 0 = ω. The continuous dependence of the solutions on initial data is stated in Theorem 3.1 below, which is a particular case of [2, Theorem 4.4]. Theorem 3.1. Let N be a boundaryless smooth manifold and g : R Ñ Rl a functional field. Assume, for any ω Ñ, the uniqueness of the maximal solution of problem 3.2. Then, given T > 0, the set D = { } ω Ñ : the maximal solution of 3.2 is defined up to T is open and the map ω D z ω T Ñ, where zω is the unique maximal solution of problem 3.2, is continuous. More generally, we will need to consider a RFDE depending on parameters of the form z t = hα, t, z t, 3.3 where h: R s R Ñ Rl is a parametrized functional field over N. The continuous dependence on data of the solutions of the initial value problems associated with 3.3 is ensured by the following consequence of Theorem 3.1. Corollary 3.2 Continuous dependence on data. Let N be a boundaryless smooth manifold and h: R s R Ñ Rl a parametrized functional field. For any α R s and ω Ñ, assume the uniqueness of the maximal solution of the problem { z t = hα, t, z t, 3.4 z 0 = ω. Then, given T > 0, the set D = { α, ω R s } Ñ : the maximal solution of 3.4 is defined up to T is open and the map α, ω D z α,ω T Ñ, where zα,ω is the unique maximal solution of problem 3.4, is continuous. Proof. Apply Theorem 3.1 to the problem { β t, z t = 0, hβt, t, z t, β0, z0 = α, ω, that can be regarded as an initial value problem of a RFDE on the ambient manifold R s N R s+l. In Theorem 3.1 and in Corollary 3.2 above the hypothesis of the uniqueness of the maximal solution of problems 3.2 and 3.4 is essential in order to make their statements meaningful. Sufficient conditions for the uniqueness are presented in Remark 3.3 below.
Global continuation of forced oscillations 9 Remark 3.3. A functional field g : R Ñ Rl is said to be compactly Lipschitz for short, c-lipschitz if, given any compact subset Q of R Ñ, there exists L 0 such that gt, γ gt, δ L γ δ for all t, γ, t, δ Q. Moreover, we will say that g is locally c-lipschitz if for any τ, ω R Ñ there exists an open neighborhood of τ, ω in which g is c-lipschitz. In spite of the fact that a locally Lipschitz map is not necessarily globally Lipschitz, one could actually show that if g is locally c-lipschitz, then it is also globally c-lipschitz. As a consequence, if g is locally Lipschitz in the second variable, then it is c-lipschitz as well. In [2] we proved that, if g is a c-lipschitz functional field, then problem 3.2 has a unique maximal solution for any ω Ñ. For a characterization of compact subsets of Ñ see, e.g., [8, Part 1, IV.6.5]. In addition to the assumptions of Corollary 3.2, assume that h is T -periodic with respect to t. We will associate with problem 3.4 a parametrized Poincaré-type T -translation operator whose fixed points are the restrictions to the interval [ T, 0] of the T -periodic solutions of 3.4. For this purpose, we need to introduce a suitable backward extension of the elements of N = C[ T, 0], N. The properties of such an extension are contained in Lemma 3.4 below, obtained in [11]. In what follows, by a T -periodic map on an interval J we mean the restriction to J of a T -periodic map defined on R. Lemma 3.4. There exist an open neighborhood U of N in N and a continuous map from U to Ñ, η η, with the following properties: 1 η is an extension of η; 2 η is T -periodic on, T ]; 3 η is T -periodic on, 0], whenever η N. Take U N as in the previous lemma. Given α R s and η U, consider the initial value problem { z t = hα, t, z t, 3.5 z 0 = η, where η is the extension of η as in Lemma 3.4. Let E = { α, η R s U : the maximal solution of 3.5 is defined up to T }. By Corollary 3.2 and Lemma 3.4 it follows that E is open in R s N. Given α, η E, denote by z α, η the maximal solution of problem 3.5 and define H: E N
10 P. Benevieri, A. Calamai, M. Furi and M. P. Pera by Hα, ηθ = z α, η θ + T, θ [ T, 0]. Observe that Hα, η is the restriction of z α, η T Ñ to the interval [ T, 0]. The following theorems regard crucial properties of the operator H. The proof of the first one is standard, while the proof of the second one is similar to that contained in [3, Lemma 3.7] and thus will be omitted. Theorem 3.5. The fixed points of Hα, correspond to the T -periodic solutions of equation 3.3 in the following sense: η U is a fixed point of Hα, if and only if it is the restriction to [ T, 0] of a T -periodic solution of 3.3 corresponding to α. Theorem 3.6. The operator H is continuous and locally compact. 4. Retarded functional motion equations Let M R k be a boundaryless smooth m-dimensional manifold and let T M = { q, v R k R k : q M, v T q M } denote the tangent bundle of M. This is a smooth 2m-dimensional manifold containing a natural copy of M via the embedding q q, 0. The natural projection of T M onto M is just the restriction to T M as domain and to M as codomain of the projection of R k R k onto the first factor. Given q M, let T q M R k denote the normal space of M at q. Since R k = T q M T q M, any vector u R k can be uniquely decomposed into the sum of the parallel or tangential component u π T q M of u at q and the normal component u ν T q M of u at q. As previously, denote by M the set BU, 0], M and let f : R M R k be a functional field over M. In this section we will consider the following parametrized retarded functional motion equation on the constraint M: x πt = λft, x t, λ 0, 4.1 where x πt stands for the parallel component of the acceleration x t R k at the point xt. In the case when the active force λf is identically zero, equation 4.1 reduces to the so-called inertial equation x πt = 0, and one obtains the geodesics on M as solutions. Given any q, v T M, there exists one and only one maximal geodesic with initial position q and initial velocity v T q M. This geodesic can be regarded as a curve in R k, and if v 0 it has a curvature at q that we denote kq, v. It is convenient to put kq, v = 0 whenever v = 0. Observe that, given q, v T M and a real number s 0, one has kq, v = kq, sv.
Global continuation of forced oscillations 11 This is because the two geodesics with initial position q and initial velocities v and sv have the same image. Obviously, kq, v depends continuously on q and v T q M with v = 1. Therefore, given a compact subset C of M, there exists a constant K such that kq, v K for all q, v T M with q C. It is known that equation 4.1 can be equivalently written as x t = rxt, x t + λft, x t, λ 0, 4.2 where r : T M R k is a smooth map the so-called reactive force or inertial reaction satisfying the following properties: a rq, v T q M for any q, v T M; b rq, v is quadratic in v and its norm equals kq, v v 2 ; c given q, v T M, rq, v is the unique vector of R k such that v, rq, v belongs to T q,v T M; d given any C 2 curve γ : a, b M, the normal component γ ν t of γ t at γt verifies the condition γ ν t = rγt, γ t, t a, b. Now, the second order equation 4.2 can be transformed into the first order system { x t = yt, y 4.3 t = rxt, yt + λft, x t, λ 0, which is actually a first order parametrized RFDE on the tangent bundle T M, since the map λ, t, φ, ψ [0, + R T M ψ0, rφ0, ψ0 + λft, φ is a parametrized functional field over T M. Thus, recalling the notion of solution introduced in Section 3 for first order RFDEs, we get that a C 1 function x: J M, defined on an open real interval J with inf J =, is a solution of 4.1 if the pair x, x : J T M is a solution of 4.3. From now on, assume in addition that f is T -periodic in the first variable T > 0. Given λ 0, by a T -periodic solution, or forced oscillation, of equation 4.1 we mean a solution which is globally defined on R and is T - periodic. In what follows, it will be useful to consider the forced oscillations of 4.1 as elements of the space CT 1 M of the M-valued T -periodic C1 functions defined on R. This is a metric space, as a subset of the Banach space CT 1 Rk with the standard norm x = max xt + max t R t R x t. The pairs λ, x [0, + CT 1 M, with x: R M a forced oscillation of 4.1 corresponding to λ, will be called T -forced pairs of 4.1, and we denote by X the set of these pairs. Among the T -forced pairs we shall consider as trivial those of the type 0, x with x constant. Henceforth, given any q M, we will denote by q the constant map t q, t R; so that any trivial T -forced pair is of the type 0, q. In the set X we shall distinguish the subset X = { 0, q [0, + C 1 T M : q M } 4.4
12 P. Benevieri, A. Calamai, M. Furi and M. P. Pera consisting of the trivial T -forced pairs. Consequently, its complement in X, X\X, will be called the set of the nontrivial T -forced pairs. Notice that there may exist nontrivial T -forced pairs of the type 0, x, provided that x: R M is a nonconstant T -periodic geodesic of M. Once we have a set usually made up of the solutions of a given equation, a distinguished subset called the subset of the trivial elements and a convenient topology on the set, we may define the concept of bifurcation point: it is a trivial element or an alias of it with the property that any of its neighborhoods contains nontrivial elements. Thus, in our case, the alias q of a trivial T -forced pair 0, q is a bifurcation point of equation 4.1 if any neighborhood of 0, q in [0, + C 1 T M contains nontrivial T -forced pairs. Let us point out that, if q is a bifurcation point, any nontrivial T -forced pair λ, x sufficiently close to 0, q must have λ > 0 since, as well known, in a Riemannian manifold there are no nonconstant closed geodesics too close to a given point. The same fact could also be deduced from Lemma 4.1 below that will be used in proving our necessary condition for bifurcation. Lemma 4.1 See [10]. Given any periodic C 2 curve x: R M, let If Kd < 1, then d = sup { xt 1 xt 2 : t 1, t 2 R }, u = sup { x t : t R }, F = sup { x πt : t R }, K = sup { kxt, x t : t R }. u 2 F d 1 Kd. Remark 4.2. Lemma 4.1 implies that any nonconstant closed geodesic must satisfy Kd 1; since otherwise, F being zero, one would get u = 0. For example, for a nonconstant closed geodesic on a sphere in R 3 one has Kd = 2. In what follows, we are interested in the existence of branches of nontrivial T -forced pairs that, roughly speaking, emanate from a trivial pair 0, q, with q a bifurcation point of 4.1. To this end, we introduce the average force f : M R k given by fq = 1 T T 0 ft, q dt, which is clearly a vector field tangent to M. Throughout the paper, f will play a crucial role in obtaining our continuation results for 4.1. Observe that, in order to avoid the use of an extra notation, given q M, we will still denote by q the constant q-valued function defined on R or on any convenient real interval. The actual domain of q will be clear from the context in the above formula, for example, the domain of q is, 0].
Global continuation of forced oscillations 13 Theorem 4.3 below, provides a necessary condition for q M to be a bifurcation point. Namely, it asserts that any bifurcation point is contained in the following subset of X : X = = { 0, q X : fq = 0 }. 4.5 In what follows, to avoid misleading overlaps with the standard notation x t used in the RFDEs context, we will denote by x n the nth term of a sequence of functions in C 1 T M. Theorem 4.3. Let M R k be a boundaryless smooth manifold and let f : R M R k be a functional field on M, T -periodic in the first variable. If q M is a bifurcation point of 4.1, then fq = 0. Consequently, the set X = { 0, q X : fq 0 } 4.6 is a relatively open subset of the set X of the T -forced pairs. Proof. Let q M be a bifurcation point of 4.1. Thus, there exists a sequence {λ n, x n } of nontrivial T -forced pairs of 4.1 converging to 0, q in [0, + C 1 T M. Since any λ n, x n is a nontrivial pair, as previously observed we have λ n > 0 for n N sufficiently large. We also have x n t q uniformly on R, so that there exists a compact subset of M containing x n t for all t R and n N. Hence, there exists a positive constant K such that rx n t, x n t = k x n t, x n t x n t 2 K x n t 2. By integrating the equality on [0, T ], we get 0 = x n T x n 0 = which implies x n t = r x n t, x n t + λ n ft, x n t T 0 r x n t, x n t T dt + λ n ft, x n t dt T T λ n ft, x n t dt K x n t 2 dt. 4.7 0 Now, observe that the sequence of curves t t, x n t R M converges uniformly to t t, q for t [0, T ]. Hence, because of Lemma 2.2, ft, x n t ft, q uniformly for t [0, T ] and, thus, there exists F > 0 such that ft, x n t F. Let d n = max { x n t 1 x n t 2 : t 1, t 2 [0, T ] }. Since d n 0 as n +, one has Kd n < 1 for n sufficiently large. Therefore, by applying Lemma 4.1, we obtain x n t 2 F d n λ n 1 Kd n 0 0
14 P. Benevieri, A. Calamai, M. Furi and M. P. Pera for n sufficiently large and all t [0, T ]. Consequently, by 4.7, T ft, x n t dt KT F d n. 1 Kd n 0 Thus, passing to the limit, we get the necessary condition for q to be a bifurcation point: fq = 0. Finally, the last assertion follows easily from the necessary condition, taking into account that the tangent vector field f : M R k is continuous. Let now Ω be an open subset of [0, + CT 1 M. Our main result Theorem 4.4 below provides a sufficient condition for the existence of a bifurcation point q in M with 0, q Ω. More precisely, we give conditions which ensure the existence in Ω of a connected subset of nontrivial T -forced pairs of equation 4.1 an Ω-global bifurcating branch for short, whose closure in Ω is noncompact and intersects the set of trivial T -forced pairs contained in Ω. Theorem 4.4. Let M R k be a boundaryless smooth manifold, f : R M R k a functional field on M, T -periodic in the first variable and locally Lipschitz in the second one, and f : M R k the autonomous tangent vector field fq = 1 T T 0 ft, q dt. Let Ω be an open subset of [0, + C 1 T M and let j : M [0, + C1 T M be the embedding q 0, q. Assume that deg f, j 1 Ω is defined and nonzero. Then, Ω contains a connected subset of nontrivial T -forced pairs of equation 4.1 whose closure in Ω is noncompact and meets the set { 0, q Ω : fq = 0 }. The proof of Theorem 4.4 requires some preliminaries and will start later. As in Section 3, denote by T M the set C[ T, 0], T M and by T M the subset of T M of those elements φ, ψ such that φ T, ψ T = φ0, ψ0. Let U be an open neighborhood of T M in T M as in Lemma 3.4. Given λ 0 and φ, ψ U, consider the initial value problem x t = yt, y t = rxt, yt + λft, x t, 4.8 x 0 = φ, y 0 = ψ, where φ, ψ T M is the extension of φ, ψ as in Lemma 3.4.
Global continuation of forced oscillations 15 Since f is locally Lipschitz in the second variable, for any λ [0, + the uniqueness of the maximal solution of problem 4.8 is ensured recall Remark 3.3 and, because of Corollary 3.2 and Lemma 3.4, the set D = { λ, φ, ψ [0, + U : the maximal solution of 4.8 is defined up to T } 4.9 is an open subset of [0, + T M. Also observe that D 0, the slice of D at λ = 0, contains the set { q, 0 T } M : q M, which is a natural copy of M in T M. Define P : D T M by P λ, φ, ψθ = x λ, φ, ψ θ + T, y λ, φ, ψ θ + T, θ [ T, 0], where x λ, φ, ψ, y λ, φ, ψ denotes the maximal solution of problem 4.8. We observe that, given λ 0, the partial map P λ, is, in some sense, a Poincaré-type T -translation operator of system 4.3 corresponding to the frozen parameter λ. In fact, if the solution of problem 4.8 is defined up to T, P λ, φ, ψ is the backshift to the interval [ T, 0] of the restriction to [0, T ] of this solution. By Theorem 3.5 with N = T M, E = D, H = P, η = φ, ψ, α = λ, z = x, y, we get that the fixed points of P λ, correspond to the T -periodic solutions of system 4.3. More precisely, the coincidence set S = { λ, φ, ψ D : P λ, φ, ψ = φ, ψ } is the image of the set X of the T -forced pairs under the injective linear operator ρ: R CT 1 R k R C [ T, 0], R k R k given by λ, x λ, φ, ψ, where φ and ψ are the restrictions of x and x to the interval [ T, 0], respectively. As for the set X, also for S we may introduce the subsets labeled with the symbols, =, see formulas 4.4, 4.5, 4.6. Each one of these sets is the image under ρ of the analogous subset of X with the same label. Actually, as we shall see, if two subsets, one of X and one of S, have the same label, they correspond homeomorphically under ρ. Remark 4.5. The λ = 0-slice S 0 of S contains the following copy of M: { q, 0 T } M : q M, made up of the trivial fixed points of P 0,. Obviously S 0 contains as well the set G T of the nontrivial fixed points of P 0, ; that is, the elements φ, ψ T M
16 P. Benevieri, A. Calamai, M. Furi and M. P. Pera such that φ and ψ are, respectively, the restrictions to the interval [ T, 0] of a nonconstant T -periodic geodesic x: R M R k and of its derivative x : R R k. The following crucial lemma shows that the nontrivial fixed points of P 0, do not contribute in the computation of the fixed point index of the operator P λ,, provided that λ > 0 is sufficiently small. Lemma 4.6. Let V be an open subset of T M. Assume that for some δ > 0 the set [0, δ] V is contained in the domain D of the Poincaré operator P and its image under P is relatively compact in T M. If P 0, has no fixed points along V and no trivial fixed points in V, then for λ 0 sufficiently small one has ind T M P λ,, V = 0. Proof. Observe first that, because of the compactness of the restriction of P to the set [0, δ] V and by virtue of the assumption that P 0, is fixed point free along V, P λ, remains fixed point free along V also for λ > 0 sufficiently small. Taking δ smaller, if necessary, we may assume that this holds for 0 λ δ. Thus ind T M P λ,, V is defined for any λ [0, δ] and, as a consequence of the homotopy invariance, this index does not depend on λ. Therefore, it is sufficient to prove that ind T M P 0,, V = 0. To this purpose consider the equation x πt = εx t, 4.10 obtained by adding to the inertial motion equation on M a frictional force with coefficient ε 0. As was done for the definition of P, associated with equation 4.10 one may consider a Poincaré-type translation operator G: [0, + V T M. Precisely, given ε, φ, ψ [0, + V, Gε, φ, ψ is the back-shift to the interval [ T, 0] of the restriction to the interval [0, T ] of the solution of the initial value problem x t = yt, y t = rxt, yt εyt, x 0 = φ, y 0 = ψ, provided that this solution is defined up to T. Clearly, 4.10 being an ODE, the solution of the above problem depends only on the initial point φ0, ψ0 T M. We claim that the operator G is defined on the whole set [0, + V. To see this, observe first that G0, = P 0,. Consequently, G0, is defined on V, since so is P 0,. Now, the bigger is ε, the slower is the motion on M. Thus, when ε = 0, if a solution xt of 4.10 is defined up to T, the same property is a fortiori verified, when ε > 0, by the solution ut with the same
Global continuation of forced oscillations 17 initial condition x0, x 0 as xt. Actually, as one can check, for t 0 one has ut = xαt, where α: R R is the unique function such that α t = εα t, α0 = 0, α 0 = 1. This proves our claim since αt = 1 ε 1 e εt < t for t > 0. If ε > 0, the only T -periodic solutions of 4.10 are the trivial ones; that is, the constant functions q CT 1 M with q M. This can be easily deduced, for example, from the fact that if ut is a nonstationary solution of 4.10, then its kinetic energy 1 2 u t 2, when ε > 0, is strictly decreasing. Consequently, when ε > 0, the fixed points of the operator Gε, are all trivial, i.e., elements of T M of the type q, 0, with q M. Since, by assumption, the set V does not contain any one of these points, then ind T M Gε,, V = 0 for all ε > 0. Hence, recalling that G0, = P 0, and that P 0, is fixed point free along V, the homotopy invariance property of the fixed point index implies that 0 = ind T M G0,, V = ind T M P 0,, V, and the assertion follows applying the homotopy invariance to the family of operators P λ,, λ [0, δ]. The following lemma states another crucial property of the set of the T -forced pairs. Lemma 4.7. The set X of the T -forced pairs of equation 4.1 is closed in [0, + C 1 T M. Proof. Let {λ n, x n } be a sequence in X that converges to an element λ, x in [0, + CT 1 M. Observe that the sequence of functions t t, x n t R M converges uniformly to t t, x t for t R. Hence, because of Lemma 2.2 and the T -periodicity of the above functions, λ n ft, x n t λft, x t uniformly for t R. Analogously, rx n t, x n t converges uniformly to r xt, x t. Thus x n t converges uniformly in R to αt := r xt, x t + λft, x t. Therefore, taking into account that x n t converges to x t for t R, we get that x t is differentiable and its derivative, x t, coincides with αt. We turn now our attention to the following two-parameter family of retarded functional motion equations on M: x πt = λ 1 µft, x t + µ fxt, λ 0, µ [0, 1]. 4.11 Notice that for µ = 0 this equation coincides with 4.1, while for µ = 1 one gets a second order autonomous ODE on M.
18 P. Benevieri, A. Calamai, M. Furi and M. P. Pera The triples λ, x, µ [0, + C 1 T M [0, 1], with x: R M a forced oscillation of 4.11 corresponding to λ and µ, will be called T -forced triples of 4.11. Denote by Ξ the set of these triples and by Ξ = { 0, q, µ : q M, µ [0, 1] } the distinguished subset of Ξ of the trivial T -forced triples. These sets are metric spaces, as subsets of the Banach space R C 1 T Rk R with norm λ, x, µ = λ + max xt + max t R t R x t + µ. 4.12 Notice that the slices at µ = 0 of Ξ and Ξ are, respectively, the set X of the T -forced pairs of equation 4.1 and its subset X of the trivial ones. The two-parameter equation 4.11 can be transformed into the following family of first order systems on T M: { x t = yt, y t = rxt, yt + λ 1 µft, x t + µ fxt, λ 0, µ [0, 1]. 4.13 Given λ 0, µ [0, 1] and φ, ψ U, with U as above, consider the initial value problem x t = yt, y t = rxt, yt + λ 1 µft, x t + µ fxt, 4.14 x 0 = φ, y 0 = ψ, where φ, ψ T M is the extension of φ, ψ as in Lemma 3.4. Since f is locally Lipschitz in the second variable, then it is easy to see that f is locally Lipschitz as well. Hence, for any λ [0, + and µ [0, 1], the uniqueness of the maximal solution of problem 4.14 is ensured recall Remark 3.3 and, because of Corollary 3.2 and Lemma 3.4, the set E = { λ, φ, ψ, µ [0, + U [0, 1] : the maximal solution of 4.14 is defined up to T } is an open subset of [0, + T M [0, 1]. Also observe that E 0, the slice of E at λ = 0, contains the set { q, 0, µ T } M [0, 1] : q M, µ [0, 1]. Define H : E T M by Hλ, φ, ψ, µθ = x λ, φ, ψ,µ θ + T, y λ, φ, ψ,µ θ + T, θ [ T, 0], where x λ, φ, ψ,µ λ, φ, ψ,µ, y
Global continuation of forced oscillations 19 denotes the maximal solution of problem 4.14. Notice that, given λ 0 and µ [0, 1], the map Hλ,, µ is a Poincaré-type T -translation operator of system 4.13 corresponding to the frozen parameters λ and µ. In fact, if the solution of problem 4.14 is defined up to T, Hλ, φ, ψ, µ is the back-shift to the interval [ T, 0] of the restriction to [0, T ] of this solution. By Theorem 3.5, with N = T M, E = E, H = H, η = φ, ψ, α = λ, µ, z = x, y, we get that the fixed points of Hλ,, µ correspond to the T -periodic solutions of system 4.13. More precisely, the coincidence set Σ = { λ, φ, ψ, µ E : Hλ, φ, ψ, µ = φ, ψ } is the image of the set Ξ of the T -forced triples under the injective linear operator σ : R C 1 T R k R R C [ T, 0], R k R k R given by λ, x, µ λ, φ, ψ, µ, where φ and ψ are the restrictions of x and x to the interval [ T, 0], respectively. We have already introduced a norm in the domain of σ see 4.12. As regards the codomain R C[ T, 0], R k R k R, defining λ, φ, ψ, µ = λ + max φt + t [ T,0] max t [ T,0] ψt + µ, the map σ establishes an isometry between the two metric spaces Ξ and Σ. Anyhow, what is important in what follows is that σ maps Ξ onto Σ, homeomorphically. Observe that the µ = 0-slices of Ξ and Σ are the sets X and S, that have been already introduced and denoted, respectively, with the Latin letters corresponding to Ξ and Σ. Obviously X and S correspond, homeomorphically, under the partial map ρ = σ,, 0 of σ obtained by freezing µ = 0. Roughly speaking, the elements of S are the initial conditions of all the T -forced pairs of 4.1. Finally, Theorem 3.6 implies that the operator H is locally compact. Therefore, taking into account that the projection λ, φ, ψ, µ φ, ψ is a locally proper map, the coincidence set Σ is locally compact. Consequently, so is the set Ξ of the T -forced triples, being homeomorphic to Σ. Hence the set X of the T -forced pairs is locally compact, as a slice of a locally compact space. In fact, it is homeomorphic to the closed subset X {0} of Ξ. Proof of Theorem 4.4. Let, as before, X denote the set of all the T -forced pairs of equation 4.1. Recall that this set is closed in [0, + CT 1 M, as stated in Lemma 4.7, and locally compact, as observed above. As in 4.4, denote by X the distinguished subset of X of the trivial T -forced pairs, and consider the subset X=, defined in 4.5, of the elements of X that satisfy the necessary condition for bifurcation. According to this notation, to prove Theorem 4.4 we need to show that there exists a connected subset of X\X Ω whose closure in Ω is noncompact and intersects X=.
20 P. Benevieri, A. Calamai, M. Furi and M. P. Pera Now, define ˇX = X\X X= and observe that this set is closed in [0, + CT 1 M. In fact, ˇX coincides with X\X, where X, defined in 4.6, is a relatively open subset of X because of Theorem 4.3. Thus, the closure in ˇX Ω of a subset of ˇX Ω is the same as in Ω. Consequently, to prove the assertion of Theorem 4.4 it is enough to show that there exists a connected subset of X\X Ω whose closure in ˇX Ω is noncompact and intersects X=. Now, since X is locally compact, the closed subset ˇX of X is locally compact as well. Hence, so is the open subset ˇX Ω of ˇX. Put Y = ˇX Ω and observe that the subset K = X= Ω of Y is compact because of the assumption that deg f, j 1 Ω is well defined. Therefore, according to Lemma 2.1, to prove Theorem 4.4 it is enough to show that any compact subset of Y containing K has nonempty boundary in Y. By contradiction, suppose that there exists a compact subset C of Y containing K with empty boundary in Y. This implies that C is relatively open in Y and, therefore, in ˇX as well. Consequently, the image C of C under the isometry ρ is a compact, relatively open subset of Š = S\S S = = ρ ˇX. Thus, there exists an open subset W of [0, + T M such that C = Š W and Š W =. Notice that the coincidence set S is contained in the open set D defined in 4.9. Further, observe that D = { λ, φ, ψ [0, + U : λ, φ, ψ, 0 E } and P λ, φ, ψ = Hλ, φ, ψ, 0. Therefore, we may assume that the closure W of W is contained in D, so that the Poincaré-type T -translation operator P : D T M as in 4.9 is well defined on W. We recall that, due to Theorem 3.5, any fixed point of P λ, is a starting element of a T -periodic solution of 4.1. Observe also that the set of fixed points of P λ, in the slice W λ of W is the slice C λ of C. Since C is compact, for any λ > 0 the fixed point index ind T M P λ,, Wλ is defined and independent of λ because of the generalized homotopy invariance. Now, C λ being empty for some λ > 0, we get ind T M P λ,, Wλ = 0 4.15 for all λ > 0. Therefore, the assertion will follow, by contradiction, if we show that ind T M P λ,, Wλ 0 for some λ > 0. We will prove that this holds for λ > 0 sufficiently small. To this purpose, it is convenient to assume that W is mapped by P into a
Global continuation of forced oscillations 21 relatively compact subset of T M. This is possible because of the local compactness of P and the compactness of C. Notice that ind T M P 0,, W 0 could be undefined. In fact, the set of fixed points of P 0, coincides with W 0 S 0 and, as already pointed out see Remark 4.5 the λ = 0-slice S 0 of S contains a copy of the possibly noncompact manifold M, made up of the trivial fixed points of P 0,. Recall that S 0 contains as well the set G T of the nontrivial fixed points of P 0,. Let V T M be an open neighborhood of the λ = 0-slice C 0 of C with the property that [0, λ + ] V is contained in W for some λ + > 0. The existence of such an open set is ensured by the compactness of C 0. Since the closed subset W 0 \V of T M does not contain elements of C 0, the compactness of C implies that there are no fixed points of P λ, in W λ \V for λ > 0 sufficiently small observe that, for λ > 0, the set of fixed points in W λ of P λ, coincides with C λ, even if it is not so for λ = 0. Therefore, for these values of λ, the excision property of the fixed point index implies that ind T M P λ,, Wλ = ind T M P λ,, V. 4.16 Recalling that nonconstant geodesics cannot accumulate to a given point, we may assume that V is the disjoint union of two open sets. One of them containing the set G T V of all the nontrivial fixed points of P 0, in V, and the other one, say V, containing all the remaining elements of C 0 that is, among the trivial fixed points q, 0 V, only those satisfying the necessary condition for bifurcation fq = 0. Using the additivity property of the fixed point index and applying Lemma 4.6, for λ > 0 small we get ind T M P λ,, V = ind T M P λ,, V, and from 4.16 one has ind T M P λ,, Wλ = ind T M P λ,, V 4.17 for the above values of λ. We claim that, taking both V and λ + smaller if necessary, we have ind T M P λ,, V = ind T M Hλ, 0, V for all λ 0, λ + ]. In fact, consider the set = ind T M Hλ, 1, V 4.18 O = { λ, φ, ψ [0, + U : λ, φ, ψ, µ E µ [0, 1] }, which is clearly open, due to the compactness of [0, 1]. Moreover, it contains the compact set K = {0} C 0 V = { 0, q, 0 W : fq = 0 } = ρk.
22 P. Benevieri, A. Calamai, M. Furi and M. P. Pera Therefore, if necessary taking both V and λ + smaller, we may assume that [0, λ + ] V is contained in O, so that the homotopy H λ : V [0, 1] T M, given by φ, ψ, µ Hλ, φ, ψ, µ, is well defined for any λ [0, λ + ]. Since, as already pointed out, the map H : E T M is locally compact, without loss of generality we may suppose that the image under H of the box = [0, λ + ] V [0, 1] is relatively compact in T M. This is possible due to the compactness of K and [0, 1], again taking V and λ + smaller, if necessary. Of course, after having chosen V, λ + can be taken so small that 4.17 holds for all λ 0, λ + ]. Since, in T M, H is relatively compact and is closed, the coincidence set Σ = { λ, φ, ψ, µ : Hλ, φ, ψ, µ = φ, ψ } is compact. Recalling that Σ is an open subset of Σ and that ˇΣ = Σ\Σ, we get that ˇΣ is closed in Σ and consequently compact. Therefore, so is its image ˇF under the projection of onto [0, λ + ] V. Notice that, given λ 0, λ + ], for the slice ˇF λ = { φ, ψ V : λ, φ, ψ ˇF } of ˇF one has ˇF λ = { φ, ψ V : φ, ψ is a fixed point of H λ, µ for some µ [0, 1] }. 4.19 We point out that this is not so for the λ = 0-slice of ˇF since, because of the choice of V, ˇF 0 coincides only with the set of trivial fixed points q, 0 of P 0, in V satisfying the necessary condition for bifurcation fq = 0. Since ˇF 0 does not intersect the closed set V, the compactness of ˇF implies that the same holds for any slice ˇF λ with λ > 0 sufficiently small. Taking λ + smaller, if necessary, we may assume that this happens for all λ 0, λ + ]. Now, fix any λ 0, λ + ] and observe that the homotopy invariance property of the fixed point index applies to the compact homotopy H λ : V [0, 1] T M since, because of 4.19, the condition ˇF λ V = means that H λ φ, ψ, µ φ, ψ for all φ, ψ, µ V [0, 1]. Thus, ind T M Hλ, 0, V = ind T M Hλ, 1, V and, recalling that H λ, 0 coincides with the Poincaré operator P λ,, we get that equality 4.18 holds, as claimed. Consequently, taking into account 4.17, we obtain ind T M P λ,, Wλ = ind T M Hλ, 1, V
Global continuation of forced oscillations 23 for all λ 0, λ + ]. Now, as we already pointed out, the assertion of the theorem follows, by contradiction, if we show that ind T M P λ,, Wλ 0 for some λ > 0. To this purpose, we shall compute ind T M H λ, 1, V for λ > 0 small. Let V = { q, v T M : q, v V } and let Q: [0, + V T M be the classical finite-dimensional Poincaré T -translation operator Qλ, q, v = x λ,q,v T, y λ,q,v T, where x λ,q,v, y λ,q,v denotes the maximal solution of the undelayed system x t = yt, y t = rxt, yt + λ fxt, λ 0, x0 = q, y0 = v. An argument based on the commutativity property of the fixed point index analogous to that used in the proof of Lemma 3.8 Step 5 of [3] shows that, for small values of λ, ind T M Qλ,, V is defined and ind T M Hλ, 1, V = indt M Qλ,, V. Now, as shown in [10, Lemma 2.2], for λ > 0 sufficiently small we get ind T M Qλ,, V = 1 m deg f, i 1 V, where m = dim M and i: M T M is the embedding q q, 0. Finally, by the excision property of the degree, we obtain deg f, i 1 V = deg f, j 1 Ω that, by assumption, is nonzero. Consequently, by taking into account 4.15 and the assumption deg f, j 1 Ω 0, we get a contradiction. This completes the proof. Remark 4.8 On the meaning of global bifurcating branch. In addition to the hypotheses of Theorem 4.4, assume that f sends bounded subsets of R M into bounded subsets of R k, and that M is closed in R k or, more generally, that the closure Ω of Ω in [0, + CT 1 M is complete. Then a connected subset Γ of Ω as in Theorem 4.4 is either unbounded or, if bounded, its closure Γ in Ω reaches the boundary Ω of Ω. To see this, assume that Γ is bounded. Then, being fγ bounded, because of Ascoli s theorem, Γ is actually totally bounded. Thus, Γ is compact, being totally bounded and, additionally, complete since Γ is contained in Ω.